Factorization of simple modules for certain restricted two-parameter quantum groups

Abstract

We study the representations of the restricted two-parameter quantum groups of types B and G. For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive ℓth root of unity, the factorization of simple ${\mathfrak{u}}_{{\theta }^y,{\theta }^z}(\mathfrak{g})$-modules is possible, if and only if (2(y − z), ℓ) = 1 for $\mathfrak{g}=\mathfrak{s}{\mathfrak{o}}_{2n+1}$; (3(y − z), ℓ) = 1 for $\mathfrak{g}$ = G₂.

Keywords

Hopf algebra / Drinfel’d double / restricted two-parameter quantum group

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